Article and program © 2007 by Theodore Zuckerman. Pen and ink caricature of piano © 1975 by Theodore Zuckerman. If you benefited from information on this web page, and in the beat rate spreadsheet described below, I hope you will please consider making a contribution to the author for the work that he put into obtaining the information and presenting it to you in a clear, comprehensible form.

Also, you can hire the author to create or contribute to your web site. You can find more information about the author here.

**Preliminary Note:** On a keyboard instrument like a piano, the thing that you push down with your finger, in order to produce a tone, we call a *digital.* Each digital is connected to a series of *action parts* which transfer your finger motion to a swinging "hammer," the last action part in the series, which strikes either one string, two strings tuned in unison, or three strings tuned in unison. The string or strings vibrate and produce a tone having a specific pitch. Most pianos have 88 digitals which produce 88 tones, each with its own distinct pitch. The digitals are numbered from 1 to 88, from left to right, like this

click to enlarge.

**Computer:** Human, working from pitch number 49, to which you have assigned a value of ** 440 ** hz, I have calculated 88 pitches and printed them in the list below. Of those, pitch number

We can extend our piano keyboard beyond the usual 88 pitches. If, in the

pitch 1 is 27.5000

pitch 2 is 29.1352

pitch 3 is 30.8677

pitch 4 is 32.7032

pitch 5 is 34.6478

pitch 6 is 36.7081

pitch 7 is 38.8909

pitch 8 is 41.2034

pitch 9 is 43.6535

pitch 10 is 46.2493

pitch 11 is 48.9994

pitch 12 is 51.9131

pitch 13 is 55.0000

pitch 14 is 58.2705

pitch 15 is 61.7354

pitch 16 is 65.4064

pitch 17 is 69.2957

pitch 18 is 73.4162

pitch 19 is 77.7817

pitch 20 is 82.4069

pitch 21 is 87.3071

pitch 22 is 92.4986

pitch 23 is 97.9989

pitch 24 is 103.826

pitch 25 is 110.000

pitch 26 is 116.541

pitch 27 is 123.471

pitch 28 is 130.813

pitch 29 is 138.591

pitch 30 is 146.832

pitch 31 is 155.563

pitch 32 is 164.814

pitch 33 is 174.614

pitch 34 is 184.997

pitch 35 is 195.998

pitch 36 is 207.652

pitch 37 is 220.000

pitch 38 is 233.082

pitch 39 is 246.942

pitch 40 is 261.626

pitch 41 is 277.183

pitch 42 is 293.665

pitch 43 is 311.127

pitch 44 is 329.628

pitch 45 is 349.228

pitch 46 is 369.994

pitch 47 is 391.995

pitch 48 is 415.305

pitch 49 is 440.000

pitch 50 is 466.164

pitch 51 is 493.883

pitch 52 is 523.251

pitch 53 is 554.365

pitch 54 is 587.330

pitch 55 is 622.254

pitch 56 is 659.255

pitch 57 is 698.456

pitch 58 is 739.989

pitch 59 is 783.991

pitch 60 is 830.609

pitch 61 is 880.000

pitch 62 is 932.328

pitch 63 is 987.767

pitch 64 is 1046.50

pitch 65 is 1108.73

pitch 66 is 1174.66

pitch 67 is 1244.51

pitch 68 is 1318.51

pitch 69 is 1396.91

pitch 70 is 1479.98

pitch 71 is 1567.98

pitch 72 is 1661.22

pitch 73 is 1760.00

pitch 74 is 1864.66

pitch 75 is 1975.53

pitch 76 is 2093.00

pitch 77 is 2217.46

pitch 78 is 2349.32

pitch 79 is 2489.02

pitch 80 is 2637.02

pitch 81 is 2793.83

pitch 82 is 2959.96

pitch 83 is 3135.96

pitch 84 is 3322.44

pitch 85 is 3520.00

pitch 86 is 3729.31

pitch 87 is 3951.07

pitch 88 is 4186.01

**Computer:** I work fast, don't I? Most likely I calculated and displayed all 88 frequencies in less than a second. You know how long this would have taken you, human, if you had had to do it with an electronic calculator and a typewriter or pen? Probably a good few hours. You know how long this would have taken you, if you had had to do it without any kind of electronic or mechanical computing device (not even a slide rule) to aid you? Probably several days of working 8 hours per day.

**Computer Programmer:** The php program I wrote, copyright 2006 by Theodore Zuckerman, calculates all the theoretical frequencies, all the theoretical pitches, not taking inharmonicity into account, on a piano tuned to equal temperament, that is, on a piano tuned so as to have each set of pitches in an octave be a 12-tone equally-tempered octave. Theoretically, each pitch is equal to the previous pitch, multiplied by the 12th root of two. On a real piano, an initial pitch near the center of the keyboard is tuned to a reference pitch, such as a tuning fork, and then the rest of the pitches are tuned to the initial pitch, and each other; the fundamentals of one string, are tuned to the upper partials of another string. That's right, aural piano tuners *listen* to and *hear* both the fundamental frequency, and the partials, of any string they are tuning. They don't need an electronic instrument to identify the partials and determine their frequency. They don't need better than normal hearing. They learn how to do this using reference tones and their *ears*. Since the upper partials on each string are actually higher than precise harmonics, real pianos exhibit *octave stretching* — pitches turn out to be higher than their theoretical equally tempered pitch, as you move up the keyboard to the right (high) side of the initial pitch, and turn out to be lower than the theoretical equally tempered pitch as you move down the keyboard to the left side of the initial pitch. Each piano has a different amount of inharmonicity, and gets a different amount of octave stretching.

This page contains a server-side php program, so you won't see the programming code in the source document for this page. The calculations are done by the php interpeter on the web server computer (which these days, may well be a "virtual" computer at my web hosting company), then just the results are sent to your computer. If you are interested in seeing a copy of the source code, please contact me. I should add that to start with one pitch, multiply it by the twelfth root of two, note the result, multiply that by the twelfth root of 2, note the result, multiply that by the twelfth root of 2, etcetera, across the entire keyboard, from the lowest note to the highest, would result in the accumulation of imprecision. To minimize that, I calculated each pitch using a formula that related it directly to pitch 49, the standard reference pitch, which is near the middle of the keyboard. This is similar to the way we tune a piano. We don't start with a 27.5 hz tuning fork, tune the lowest A on the piano, A0, pitch 1, to 27.5, and then tune all the other pitches starting from A0. Rather, we typically start by tuning pitch 49, A4, which is near the middle of the keyboard, to 440 hz, and then tune the remainder of the pitches starting from there.

Here is the sketch of a piano keyboard that I made in 1976. I've numbered the digitals from 1 to 88 and also labeled each digital with the note letter-name commonly used for it, and the pitch it produces. Since 1976 was before I owned a personal computer, I calculated the frequencies, one by one, with an electronic calculator, and wrote them in with a pencil. Then, to halt pencil-smearing, I sprayed the sheet with fixative. Rather than provide you with a color-corrected image, showing a crisp ebony-toned and ivory-toned keyboard, I decided to provide an image which allows you to see how the paper with the sketch of the digitals has yellowed with age, yellowed more than the paper that I pasted it to and wrote the frequencies on. I'm not sure whether this is an unnecessarily nasty-looking bit of business, or a charming bit of memorabilia.

Note that the digitals, in addition to being named with numbers from 1 to 88, are named with letter names, followed by an octave number from 0 to 8. We can see for example that synonyms for **note 49**, are **A49**, and **A4**. The **4** in A4 means that this A is *the A in octave 4*. There are 8 A's on an 88-note piano, A0, A1, A2, A3, A4, A5, A6, and A7.

My **Beat Rate Calculator Spreadsheet** file for Excel, which you can purchase, not only shows the fundamental frequency of every pitch from 1 to 88 but it also shows the *coincident partials* and their *beat rates*, for *just about every equally tempered interval* that a piano tuner might ever want to listen to, when perfecting their tuning — fifths, fourths, major thirds, minor thirds, major sixths, minor 6ths, and 10ths (major third plus an octave). The beat rates for intervals that consist of an interval plus an octave can be easily determined by consulting the chart for the interval without the added octave. You can see the coincident partials and beat rates for each interval *from one end of the piano to the other *(covering more than eight 12-tone equally tempered octaves), not just for an octave of intervals in a "temperament octave." (scroll down a bit to see about getting a free sample). For example you will have 84 beat rates for 84 major thirds, progressing from A0-C1 to A7-C8, not just the 9 major thirds progressing from F3-A3 to C#4-F4. The chart will show you whether each interval is expanded or contracted. And if there is some interval that you want a chart of partials and beat rates for, that the spreadsheet file doesn't already have, I will add that on, at no extra charge. Here is how to purchase: scroll down.

The spreadsheet has been confirmed to work with Excel 2003 and 2007, and in Windows XP and Windows 7. The instruction sheet opens with Microsoft Word or Open Office. If you area unsure whether it will work with your operating system, download the free sample spreadsheet below. If the free sample works, the full spreadsheet will work.

If you change the value of A49 (A4) from 440 hz to a lower or higher value, you will be able to see how all the fundamental pitches, and consequently all the coincident partials, and all their beat rates, will be changed — all 7 pages of changes (each interval has its own page). Each page has 73 to 85 intervals, 146 to 170 partials, and 73 to 85 beat rates — they are all re-calculated almost instantaneously. This could be useful if you want to tune an instrument to equal temperament, without first raising its pitch to standard pitch, that is, when doing a "pitch raise" on the piano, raising its pitch to standard pitch in steps, or if you want to tune a historic instrument that was designed to sound and work best at a lower pitch, such as A49 = 430 hz, instead of at the contemporary standard of A49 = 440 hz.

Using Excel's Goal Seek feature, you can create a "what if" scenario: what if any pitch of *any* note, or *any beat rate* of any interval, anywhere on any page, were changed? Then, (1) what would the value of A4 have had to be, in order to get that pitch, or that beat rate, and (2) what would be all the fundamentals, partials, and beat rates, for every interval, given the new value of A? For example if you were to change C52 from 523.252 to 528, what would A49 have to be, instead of 440 (the answer is roughly 444). And then you could see what *every* fundamental would be, and what *every* partial, and *every* beat rate of *every *interval, would be, given a reference tone of 444, instead of 440. Or for example if you were to change the beat rate of the Major third, F3 to A3, from a beat rate of 6.929 hz to a beat rate of 7.0 hz, you would see the pitch of A4 change from 440 hz to 444.48 hz, and at the same time, you would see what *every* fundamental would be, and what *every* partial, and *every* beat rate of *every *interval, would be, given a reference tone of 444.48, instead of 440. Almost instantaneously! Computers never cease to amaze me, doing even these simplest things that they do, like making and displaying hundreds of calculations in less than a second.

**Easy to use:** The spreadsheet has been carefully planned so that it is easy to read and understand. All the Excel formulas can be made visible so that you can see how the fundamentals, partials, and beat rates are being calculated. Many of the variables are referred to by names, in addition to being referred to by their Excel row and column identifiers. This makes the formulas easier to understand. There are complete instructions, readable and writeable in Microsoft Word or Open Office. This means you can easily add your own notes to the instructions if you want. Scroll down a bit to see how to purchase.

**A1 to C88:** As you know, in addition to checking octaves, from end to end, you can check *any* interval, below and above your "temperament octave." From A1 to C88. Or using an alternate nomenclature, from A0 to C8. The more intervals you check the more control you can exert over how far you stretch your octaves, and you can produce a more precisely stretched, better-sounding tuning — a better-sounding piano. This is why I always check progressive 3rds, 10ths, 17ths, etcetera, far beyond the temperament octave. You could check your 10ths by using the chart for Major 3rds; just find the third that ends at the same note; however I included a chart for 10ths to make things just a little easier. You can check 11ths, 12ths, and 17ths with the charts for 4ths, 5ths, and Major 3rds or 10ths. There have been a number of articles published in the Piano Technicians' Journal describing how to listen to and tune intervals below and above the temperament octave to help determine how much to stretch.

**How to get the Beat Rate Spreadsheet?** Just click the "buy now" button below and I will send you the spreadsheet in the form of an email attachment, or if you want to download it from a web page, just let me know — I'll send you an email with an address you can click on and a user name, and a second email with a password. Just unzip, and you'll have the spreadsheet and the instructions (the instructions open in Microsoft Word). Or for an extra $2.00, I'll mail the unzipped spreadsheet to you on a CD. Once I get a paypal or credit card payment I normally respond with the attachment or the download page within 48 hours, but if I am out of town, or have an emergency, occasionally I may take longer. If you are in a hurry, you may want to contact me by email, or phone (number below), to make sure I am paying attention to my email and my paypal account.

**Any questions? **Questions about the spreadsheet before you buy? You can email me or just send me a note on the form, here. If you use the form be sure to include your email address! This particular form doesn't check to see if you've included your email address or not. It just sends me whatever you have filled in, and sends you a copy. If you don't get a copy, very likely I won't get one either! Or you can also telephone me, at **(828) 348-8088, **9 am to 8 pm EST. You can find my mailing address here. Also our email address. And after you buy the Beat Rate Spreadsheet, we will be glad to provide you with technical support for using this spreadsheet. I am a piano tuner, I was a professional tuner-technician, and I love music, musical instruments, mathematics, and pianos.

You can have the entire spreadsheet for just $16.98 (USD). Pay with PayPal or credit card.

If you'd like, you can get a free sample here of the beat rate calculator spreadsheet. If you are not sure what to expect from the full spreadsheet, or if you want to confirm that the full spreadsheet will work on your computer, then download the free sample. Save the *.zip file to a directory or to your desktop, then unzip it. Open the *.xls spreadsheet file with Excel. Open the *.doc instructions with Word (write or call if you need help). Note that the free sample is a limited version of the spreadsheet. It shows the partials and beat rates for all the fifths. It shows the partials and beat rates for one octave of fourths. Also, it is missing the feature to change A49 from 440 hz, to any value you choose, and have the spreadsheet instantly recalculate all the pitches, coincident partials, and beat rates. The full spreadsheet shows *all* the fourths and also all the Major 3rds, minor 3rds, Major 6ths, minor 6ths, and 10ths.

This page created and published with assistance from Leafy Green Web Publishing